\chapter{Summary and Conclusion}
In this thesis we have studied transport properties of strongly correlated
electron systems with a special focus on the thermoelectric transport.


In chapter \ref{chap:intro} we introduced the tools of the trade, namely the
dynamical mean field theory and its use in solving problems in strongly
correlated electron systems.  We then motived the transport calculations done
in the first half of the thesis by considering the theoretical importance and
practical application of thermoelectric transport.  We then discussed the
ongoing extension of the dynamical mean field theory for dealing with real
materials.  This motivates the second half of the thesis where the computation
of transport properties of real material within the dynamical mean field
theory is considered and the details of such a computation are expounded.

In chapter \ref{chap:HubbardModel} we introduced the Hubbard model and gave a
short overview of the dynamical mean field theory.  We then discussed a
particular method for solving the dynamical mean field equations, the so
called non crossing approximation and implemented a numerical procedure for
solving those equations.  The numerical solutions were investigated and their
region of validity established.  Analytic results were obtained for the
behavior of the solution in the asymptotic high temperature regime.


In chapter \ref{chap:transport-eqs} we started with a short introduction to
the irreversible thermodynamics of electrical and thermal transport and
considered some simple thermoelectric devices in order to introduce the
thermoelectric figure of merit, its importance and its relation to material
properties.  We then proceeded to use the results of chapter 2 to compute the
transport parameters for the Hubbard model as well as some other thermodynamic
parameters for completeness.  In order to extend our investigation to
parameter regimes not accessible by the non-crossing approximation, mainly at
low temperature, we made use of the so called iterated perturbation theory to
obtain solutions of the dynamical mean field equation and compute transport
parameters in those regimes.  Analytic modeling of the transport parameters
was carried out in the high and low temperature regimes and the low
temperature results were shown to agree well with experimental data.  We then
considered the optimization of the figure of merit in strongly correlated
system and reached the conclusion that significant figure of merit is not
likely to occur in these systems.

In chapter \ref{chap:real-transport} we extended the discussion of transport
parameters to real materials and derived the lowest order expressions for the
transport parameters in terms of real tight-binding Hamiltonians and self
energies.  We considered the small scattering limit of our expressions and
showed that it agrees with the classical Boltzmann equation expression for the
transport parameters.  The numerical evaluation of the transport parameters
was then discussed in great detail and we considered the role of symmetry and
the computation of velocities.  Finally we conducted some simple tests on the
methods that have been developed.

Chapter \ref{chap:transport-nonorth} is an extension of the work in chapter 4
to systems described in terms of a non-orthogonal basis.  We commented on the
reason why non-orthogonal bases are necessary and discussed briefly the many
body theory of electrons described in non-orthogonal bases.  We then gave a
derivation of the proper extension of the many body current operator to
non-orthogonal bases.  The expressions for the transport parameters were
re-derived and finally we discussed the alternative computation of the current
operator from basic quantum theory.

In chapter \ref{chap:computing_phi} we gave an overview of the program that
was written to compute the transport parameters according to the algorithms
developed in chapters \ref{chap:real-transport} and
\ref{chap:transport-nonorth}.

Finally in chapters \ref{chap:simple-metal} and \ref{chap:latio3_revisited} we
conducted some simple test of the computational methods.

In chapter \ref{chap:simple-metal} we considered simple metals under very
simple assumptions for the functional form of the self energy.  The
computation revealed the inherent ambiguity with such an approach when there
are multiple bands that are crossing the Fermi-surface.  The thermopower
computed for the alkali metals is in fairly good agreement with experimental
data but for the noble metals and for Lithium we predict the wrong sign for
the thermopower.  This is a well known problem and was not expected to be
resolved with such a simplistic approach.

In chapter \ref{chap:latio3_revisited} we looked at a simple model for
La$_{1-x}$Sr$_{x}$TiO$_{3}$ for which the self energy can be computed.  We
discussed the behavior of the solution as a function of the interaction energy
and found indications of the Mott metal-insulator transition at a critical $U$
which agrees well with theoretical estimates \cite{Fresard:1997}.  To nail
down the energy scales in the model the data was compared to experimental
results for both photoemission and specific heat measurements.  The bandwidth
of the model was determined from photoemission to be around 5.4eV which is
consistent with the specific heat data.

The transport functions were then computed both using the method developed in
chapters chapters \ref{chap:real-transport} and \ref{chap:transport-nonorth}
as well as with exact Fourier transform techniques which are available for
this simple model.  The two methods were found to be in good agreement with
each other.  The transport parameters were then computed from the transport
functions and found to be in reasonable agreement with experiment.

In the appendices we have gathered additional details relevant for the
computations performed in the previous chapters.

Appendix \ref{app:transforms} contains some information about efficient and
numerically accurate methods for doing integral transforms.  It is also a good
source of information about B-spline interpolation methods.

Appendix \ref{app:bethe-lattice} contains a few analytic results for integrals
associated with the computation of energy and transport parameters on the
Bethe lattice.

Appendix \ref{app:fermi-liq} contains Fermi-liquid derivations of the
transport parameters at low temperatures.

Appendix \ref{app:spectral} contains details on the spectral representation of
multi-orbital Green's functions, a topic which is not well known.  We also
discussed here the time derivatives of the Green's function and their
connections to response functions of creation and destruction operators.

In appendix \ref{app:atm} we have collected together the basic equations for
the regular tetrahedron method and rewritten the equations on a simpler and
hopefully more stable form.

In appendix \ref{app:ssdd} we extended the regular tetrahedron method to
integrands with a double pole.

In appendix \ref{chap:osod} we extended the regular tetrahedron method to
integrands with two non-degenerate single poles and derived the formulas for
the weight factors that can easily be computed analytically.  The case for the
most general type of tetrahedron still resists an analytical solution but an
approximate method was proposed.

Appendix \ref{app:dilog} contains notes on an integral that comes up in the
analytic solution obtained in appendix \ref{chap:osod}.

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